Ac33155161

Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.155-161
155 | P a g e
Shape Control And Optimization Using Cantilever Beam
Hitesh R. Patel*, J. R. Mevada**
*Student, M.Tech, (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva-
384012,
** Associate Prof., (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva-
384012,
Abstract
In this study, Analytical work is carried out for
static shape control of cantilever beam structure
with a use of laminated piezoelectric actuator
(LPA). The mathematical modeling of beam
element covered with LPA based on Timoshenko
beam element theory and linear theory of
piezoelectricity has been used. This work shows
how number of actuators, actuator size, actuator
location on beam and control voltage are
depended on the desired shape of beam. Initial
condition of beam is taken as horizontal position
and three higher order polynomial curves are
taken as desired shapes for beam to achieve.
Here error between desired shape and achieved
shape is taken as an objective to minimize, size,
location and control voltage of actuators taken as
variables. Genetic Algorithm for calculating
optimum values of all variables is carried out
using Matlab tool
Keywords – Shape control, Genetic Optimization,
Cantilever beam,
1. Introduction
Now a day due to requirement of highly
precise structure works considerable attention has
been carried out behind shape control of structure.
In some application areas like reflecting surfaces of
space antennas, aerodynamic surfaces of aircraft
wings, hydrodynamic surfaces of submarines and
ships where highly precise surfaces required.
Recently, researcher giving considerable attention
on developing advanced structures having integrated
control and self monitoring capability. Such
structures are called smart structure. Using direct
and converse effect of piezoelectric materials for
sensing and control of structures, a considerable
analytical and computational modeling works for
smart structures have been reported in the review
paper of Sunar and Rao [10], most of the past work
carried out for the control of vibration
characteristics of structures, but on shape control
side fewer work is carried out. A review paper by H.
Irschik [4] on static and dynamic shape control of
structure by piezoelectric actuation show work
carried out for shape control and relevant
application of shape control. Agrawal and Trensor
[1] show in his work analytical and experimental
results for optimal location for Piezoceramic
actuators on beam here size of Piezoceramic
actuators are not varied only optimization carried
out for optimal location of actuators and the desired
shape of the curve is taken as parabola which is a
second order polynomial curve. One more
conference report is made available by Rui Ribeiro
and Suzana Da Mota Silva [8] for the optimal design
and control of adaptive structure using genetic
optimization algorithm. They have been carried
shape control for beam in two different boundary
conditions like clamped - free and clamped –
clamped on both the end of beam and carried
optimization for location of piezo electric actuators
and sensors. Also in paper on the application of
genetic algorithm for shape control with piezo
electric patches and also show comparison with
experimental data by S daMota Silva and R Ribeiro
[9]. Paper presented by author Osama J. Aldraihem
[7] shows analytical results for optimal size and
location of piezoelectric actuator on beam for
various boundary conditions of beam at here desired
shape of beam is considered as horizontal and single
pair of actuator is used for his work. In the paper of
E.P. Hadjigeorgiou, G.E. Stavroulakis, C.V.
Massalas [3] work carried out for shape control of
beam using piezo electric actuator. Here all the
mathematical modeling is based on the Timoshenko
beam theory. They have been investigated as
placement of actuator near the fixed end of beam
And optimization carried out for the optimal voltage
of actuators. If control voltage is goes higher then
actuators get damaged so for minimization of
control voltage and increase actuation force of
actuator, layers of laminated piezoelectric actuators
(LPA) are mounted one over another so actuation
energy of patches are added and with lower voltage
we can get higher actuation force. This concept is
used by author Y Yu, X N Zhang and S L Xie [11]
in this work carried out for shape control of
cantilever beam and optimization done for optimum
control voltage.
From all above work carried out earlier by
different authors are limited to vary only position of
actuators on the beam. Here in this work number of
actuators, size and location of actuators are going to
be varied, and also three types of desired shape of
beam having higher order rather than parabolic
shape is to be considered to control.

156 | P a g e
2. Mathematical Modeling of Beam
2.1 governing equation of beam
Mathematical modeling carried out based
on Timoshenko beam element theory and linear
theory of piezoelectricity.
Figure 1. Beam with surface bonded LPA [11]
Considering Cartesian coordinate system as
shown in Fig. 1, here analysis is restricted in x-z
plane only so displacements in all three directions
(using Timoshenko beam theory) are as below [3].
Where 𝜔 is transverse displacement of point on
centroidal axis and 𝜓 is the rotation of beam cross
section about positive y axis.
𝑢1 𝑥, 𝑦, 𝑧, 𝑡 ≈ 𝑧𝜓 𝑥, 𝑡 , (1)
𝑢2 𝑥, 𝑦, 𝑧, 𝑡 ≈ 0 , (2)
𝑢3 𝑥, 𝑦, 𝑧, 𝑡 ≈ 𝜔 𝑥, 𝑡 , (3)
Nonzero strain component of beam using above
equations are as below:[3]
𝜀 𝑥 = 𝑧
𝜕𝜓
𝜕𝑥
, 𝛾𝑥𝑧 = 𝜓 +
𝜕𝜔
𝜕𝑥
, (4)
The linear piezoelectric coupling between elastic
field and electric field – no thermal effect is
considered are as following [5].
𝜎 = 𝑄 𝜀 − 𝑒 𝑇
𝐸 , (5)
𝐷 = 𝑒 𝜀 + 𝜉 {𝐸}, (6)
The equation of laminated beam is derived with the
use of Hamilton principle as below.
𝛿 𝐻 − 𝑊𝑒 𝑑𝑡 = 0
𝑡2
𝑡1
(7)
Where (.) denote first variation operator, T kinetic
energy of beam, H enthalpy of beam with laminated
piezoelectric actuator and We external work done.
Now electric enthalpy of beam [5,6] by using above
equations form (4) to (6) as
𝐻 =
1
2
𝜀 𝑇
𝜎 𝑑𝑉
𝑉 𝑏
+
1
2
𝜀 𝑇
𝜎 − 𝐸 𝑇
𝐷 𝑑𝑉
𝑉𝑝
=
1
2
𝜀 𝑇
[𝑄] 𝜀 𝑑𝑉
𝑉 𝑏
+
1
2
𝜀 𝑇
𝑄 𝜀 − 𝜀 𝑇
𝑒 𝑇
𝐸
𝑉𝑝
− 𝐸 𝑇
𝑒 𝜀 − 𝐸 𝑇
𝜉 𝐸 𝑑𝑉
==
1
2
𝐸𝐼
𝜕𝜓
𝜕𝑥
2
+
1
2
𝐺𝐴 𝜓 +
𝜕𝜔
𝜕𝑥
2𝐿 𝑒
0
− 𝑀 𝑒𝑙
𝜕𝜓
𝜕𝑥
− 𝑄 𝑒𝑙
𝜓 +
𝜕𝜔
𝜕𝑥
𝑑𝑥
−
1
2
𝐸𝑥
2
𝜉11
𝑠 𝑝
𝐿 𝑒
0
+ 𝐸𝑧
2
𝜉33 𝑑𝑠𝑑𝑥
(8)
Where,
(𝐸𝐼) = 𝑧2
𝑄11 𝑑𝑠 + 𝑧2
𝑄11𝑝 𝑑𝑠
𝑠 𝑝
,
𝑠 𝑏
𝐺𝐴 = 𝑘 𝑄55 𝑑𝑠 + 𝑄55𝑝 𝑑𝑠
𝑠 𝑝𝑠 𝑏
𝑀 𝑒𝑙
= 𝑧𝑒31 𝐸𝑧 𝑑𝑠
𝑠 𝑝
,
𝑄 𝑒𝑙
= 𝑒15 𝐸𝑥 𝑑𝑠 = 0
𝑠 𝑝
,
Here Qel
=0 because electric field intensity in x
direction is neglected and also value of k is taken as
5/6 [3].
Finally the work of external force is given by
𝑊𝑒 = 𝑞𝜔 + 𝑚𝜓 𝑑𝑥,
𝐿 𝑒
0
(9)
2.2 Finite element formulation of beam
Considering a beam element of length Le having two
degree of freedom per node one in transverse
direction W1 (or W2) and other is rotation degree of
freedom 𝜓1 (or 𝜓2). As shown in Fig 2.
Figure 2. Beam element
The array of nodal displacement is defined as
𝑿 𝑒
= 𝜔1 𝜓1 𝜔2 𝜓2
𝑇 (10)
On the basis of Timoshenko beam element
theory the cubic and quadratic Lagrangian
polynomial are used for transverse and rotation
displacement where the polynomials are made
interdependent by requiring them to satisfy the two
homogeneous differential equations associated with
Timoshenko’s beam theory. The displacement and
rotation of beam element can be expressed as
𝜔
𝜓 =
𝑁 𝜔
𝑁 𝜓
𝑿 𝑒 (11)
𝜀 𝑥
𝛾𝑥𝑧
=
0 𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑥
1
𝜔
𝜓 =
𝐵𝑢
𝐵 𝜓
𝑿 𝑒 (12)

157 | P a g e
Where, [Nω] and [Nψ] are cubic and quadratic shape
functions of tiemoshenko beam element [11].
Electric potential inside element at any arbitrary
position as
𝑉 = 1 −
𝑥
𝐿 𝑒
𝑥
𝐿 𝑒
𝑉1 𝑉2
𝑇
= 𝑁 𝑉 𝑉𝑒 (13)
And also electric field intensity Ez can be expressed
as
𝐸𝑧 = −
1
𝑛𝑡 𝑝
1 −
𝑥
𝐿 𝑒
𝑥
𝐿 𝑒
𝑉1 𝑉2
𝑇
= 𝐵𝑉 𝑉𝑒
(14)
Substituting equations (10) – (14) in equations (8)
and (9) and then substituting them in to equation (7)
the equation for shape control is obtained as bellow
[11].
𝐾 𝑒
𝑿 𝑒
= 𝐹 𝑒
+ 𝐹𝑒𝑙
𝑒 (15)
𝐾 𝑒
= 𝐾𝑢𝑢
𝑒
+ 𝐾𝑢𝑉
𝑒
𝐾𝑉𝑉
𝑒 −1
𝐾𝑢𝑉
𝑒 𝑇 (16)
Where all matrices can be express as bellow
𝐾𝑢𝑢
𝑒
=
𝜕 𝑁 𝜓
𝜕𝑥
𝑁 𝜓 +
𝜕 𝑁 𝜔
𝜕𝑥
𝐿 𝑒
0
×
𝐸𝐼 0
0 𝐺𝐴
×
𝜕 𝑁 𝜓
𝜕𝑥
𝑁 𝜓 +
𝜕 𝑁 𝜔
𝜕𝑥
𝑑𝑥,
(17)
𝐾𝑢𝑉
𝑒
= 𝐵𝑢
𝑇
𝑒31 𝐵𝑉 𝑑𝑠𝑑𝑥
𝑠 𝑝
𝐿 𝑒
0
, (18)
𝐾𝑉𝑉
𝑒
= 𝐵𝑉
𝑇
𝜉33 𝐵𝑉 𝑑𝑠𝑑𝑥,
𝑠 𝑝
𝐿 𝑒
0
(19)
To obtain stiffness matrix of beam without
coverage of laminated piezoelectric actuator put
tp=0 in equation of 𝐾𝑢𝑢
𝑒
.
And the external force matrix and electric force
matrix are as follow,
𝐹 𝑒
= 𝑁 𝜔 𝑁 𝜓
𝑞
𝑚
𝑑𝑥,
𝐿 𝑒
0
(20)
𝐹𝑒𝑙
𝑒
=
𝜕 𝑁 𝜓
𝜕𝑥
𝑁 𝜓 +
𝜕 𝑁 𝜔
𝜕𝑥
𝑀 𝑒𝑙
𝑄 𝑒𝑙 𝑑𝑥
𝐿 𝑒
0
,
(21)
3. Shape control and optimization using
genetic algorithm
Here, study carried out for static shape
control, so transverse displacement vector X is time
independent, in above equation (15) there is not any
time dependent vector so it is directly used in static
shape control problem. With fixed time independent
values of electric voltage at various location of
actuator one can control shape of beam.
3.1 Material properties of composite cantilever
beam and Piezoceramic actuator
Material selected for cantilever beam is
graphite epoxy composite because having lower
density and lower thermal coefficient of expansion,
which will minimize deflection due to temperature
rise and also due to self weight, and material
selected for actuator is PZT G1195N, properties of
both the material are specified in Table 1.[3]
Here size of beam is considered as 300 mm long,
40 mm width and 9.6 mm thick, also width and
thickness of LPA layer is considered as 40 mm and
0.2 mm respectively. Here length of LPA is going to
be varied.
3.2 Shape control
Figure 3. Cantilever beam with surface bonded LPA
Layouts of beam with laminated piezoelectric
actuators (LPA) are as shown in below Fig 3. As
shown in below figure beam divided in to 30 finite
elements having 10 mm elemental length. And one
LPA covers several elements of beam, equal
amplitude voltage with an opposite sign are
provided to upper and lower LPA, in generally says
for upward displacement of beam upper LPA need
negative voltage and lower LPA need positive
voltage.
Table 1. Material properties of base beam and LPA
Properties Symbol
LPA material
PZT G1195N
Graphite epoxy composite
material T300/976
Young modulus (GPa) E11 63 150
Poisson Ratio 𝜈12 0.3 0.3
Shear modulus (GPa) G12 24.2 7.1
Density (Kg/m3
) 𝜌 7600 1600
Piezoelectric constant (C/m2
) e13 17.584
Electric permittivity (F/m)
𝜉13 15.3 x 10-9
𝜉15 15.0 x 10-9

158 | P a g e
For shape control work initial condition of beam
is taken as horizontal beam. Here no other effect is
considered like gravity or thermal expansion of
beam. And for desired shape of beam is take as three
higher order polynomial curves having different
curvatures as shown in figure 5.
Figure 4. Desired shape of beam
Here error between desired shape and achieved
shape is to be considered as an objective function to
minimize and in this study number, size, location
and Voltage provided to the LPA are considered as
design variables. The error function used in this
study as under[8].
𝐸𝑟𝑟𝑜𝑟 = 𝛾𝑖 − 𝑞𝑖
2
𝑛
𝑖=1
(22)
Where 𝛾i is the pre-defined displacement at the ith
node and qi the achieved displacement.
3.3.1 Optimization using Genetic algorithm
For varying no of actuators on the beam here
optimization is carried out in four cases as bellow
table.
Table 2. Design variable for different cases.
Case Description
1 Using single LPA
2 Using two LPA’s
3 Using three LPA’s
4 Using four LPA’s
For Genetic algorithm population is set as 100
for all three cases and max. Number of generation is
allowed as 1000.Also algorithm having stochastic in
nature. Here crossover function is selected as
heuristic in nature. Upper Control limit for voltage
is specified as 400 V and lower limit is specified as
400 V in reverse polarity (for reverse polarity
voltage shown as minus (-) sign), minimum length
of LPA is considered as 30 mm because as length
decreased control force (Actuation force) of LPA on
beam is also decreased. And maximum length
provided as full length of beam, for empty length
upper limit is considered as full length of beam and
lower limit is considered as zero. Constraint
equation is become as the summation of all
Actuators length and Empty length of beam is less
than or equal to 300 mm.
3.3.2 Optimization for first desired shape of
beam.
Results obtained after optimization for first desired
shape are shown in bar chart as bellow. Here voltage
shown in negative direction means it is in reverse
polarity.
Figure 5. Optimization result for first shape

159 | P a g e
From above results we can say that To achieving
such desired shape one LPA is enough for optimal
control of shape, further more if number of actuators
increased we can obtain better fitness means better
control of shape, also it is required to place actuators
at higher strain area on beam, in this case higher
strain area is near the fixed end of beam.
3.3.3 Optimization for second desired shape of
beam
Results obtained after optimization for second type
of desired shape are as under.
For optimal control of such kind of desired shape
we need minimum 2 numbers of actuators, from
above results to increase number of actuators for
control of such a kind of shape is meaningless, also
there is two strain concentrated area in desired shape
of beam one at fixed end and other at the middle
position were beam is getting to deflect form lower
to upper position. Here form above results we can
also say that it is necessary to cover strain
concentrated area on beam by actuator for optimally
control the shape.
3.3.4 Optimization for third desired shape of
beam.
Results obtained after optimization for third type of
desired shape as following.
From above result we can say that it is not
possible to achieve third kind of shape using two
LPA, also further using three and four LPA and
obtained results as show in below figure 10.
Figure 6. Optimization result for second shape
Figure 7. Optimization result using one and two patches for third shape

160 | P a g e
Figure 8. Optimization result using three and four patches for third shape
Figure 9. Optimization result using three LPA (Varying number of layers) for third curve
In this third type of shape for cantilever beam there
are three strain concentrated area on the beam at
various location. From the above results better
control of shape obtained as compared to use of one
and two LPA, but voltage limit reached for LPA so
for better control of shape here multiple layers of
LPA on one over another is used, further
optimization being carried out for three LPA having
multiple layers. And results obtained from that are as
below shown in figure 11.
For Optimal shape control of third kind three
LPA are enough, there is no need to increase LPA
furthermore, there are three strain concentrated area
on the beam, also from above charts there are two
areas on beam where strain concentration is zero and
so there is no need to place LPA over there.

161 | P a g e
4 Conclusion
Here mathematical modeling of cantilever
beam based on Timoshenko beam element theory.
To minimize control cost of any structure it is
required to optimize voltage, location and size of
piezoelectric patch on the structure. Optimization is
carried out for the error minimization between
desired and actual shape with the use of genetic
optimization algorithm for different three types of
higher order polynomial curves. Here number of
actuators, voltage applied to the actuator, size of
actuator and location of actuator on beam carried as
a variable.
 Number of actuators, size of actuators, location
of actuators and control voltage provided to the
actuators are depending on the curvature of
desired shape of beam.
 For optimum shape control, Minimum number
of actuator required to control the shape of
beam is greater than or equal to number of strain
concentrated area formed during achieving
desired shape. As number of actuators increased
we get better shape control. But if number of
actuators less than required then shape control
not possible.
 For better shape control it is desired to cover
higher strain concentrated area by actuators and
there is no need to cover that portion of the
beam where strain concentration is remain zero
while achieving desired shape.
 In case of cantilever beam strain concentrated
area is near the fixed end of beam so it is
essential to cover fixed end of beam with
actuator for better shape control. There is no
need to place actuators at the free end side of
beam.
References
[1] Brij N Agrawal and Kirk E Treanor, Shape
control of a beam using piezoelectric
actuators, Smart Mater. Struct. 8 (1999)
729–740. Printed in the UK,
[2] Crawley E F and Anderson E H 1985
Detailed models of piezoelectric actuation
of beams J. Intell. Mater. Syst. 1 4–25
[3] E.P. Hadjigeorgioua*, G.E. Stavroulakisb,
C.V. Massalasa, Shape control and damage
identification of beams using piezoelectric
actuation and genetic optimization,
International Journal of Engineering
Science 44 (2006) 409–421,
[4] H. Irschik, A review on static and dynamic
shape control of structures using
piezoelectric actuation, Comput. Mech. 26
(2002) 115–128.
[5] H.F. Tiersten, Linear Piezoelectric Plate
Vibration, Plenum Press, New York, 1969.
of adaptive structures, CMS Conference
Report, 11th May 2000
[6] H.S. Tzou, Piezoelectric Shells, Kluwer
Academic Publishers, The Netherlands,
1993.
[7] Osama J. Aldraihem, Optimal Size and
Location of Piezoelectric Actuator/Sensors:
Practical Considerations, journal of
guidance, control, and dynamics Vol. 23,
No. 3, May–June 2000,
[8] Rui Ribeiro and Suzana Da Mota Silva,
Genetic algorithms for optimal design and
control of adaptive structures, CMS
Conference Report, 11th May 2000
[9] S daMota Silva1, RRibeiro2, and
JMMonteiro2, The application of genetic
algorithms for shape control with
piezoelectric patches — an experimental
comparison, Smart Mater. Struct. 13 (2004)
220–226,
[10] Sunar M and Rao S 1999 Recent advances
in sensing and control of flexible structures
via piezoelectric materials technology Appl.
Mech. Rev. 52 1–16
[11] Y Yu, X N Zhang1 and S L Xie, Optimal
shape control of a beam using piezoelectric
actuators with low control voltage, Smart
Mater. Struct. 18 (2009) 095006 (15pp),

Ac33155161

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (15)

Destaque

Destaque (20)

Semelhante a Ac33155161

Semelhante a Ac33155161 (20)

Último

Último (20)

Ac33155161